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<h3 class="heading"><span class="type">Paragraph</span></h3>
<p>The solution is</p>
<div class="displaymath process-math" data-contains-math-knowls="./knowl/eq1_8.html">
\begin{equation*}
\label{eq1_8}
S=S_0 e^{rt}+\frac{k}{r} e^{rt}(1-e^{-rt}).
\end{equation*}
</div>
<p class="continuation">Suppose <span class="process-math">\(S_0=0\text{,}\)</span> then from (<a href="" class="xref" data-knowl="./knowl/eq1_8.html" title="Equation 1.4.2">(1.4.2)</a>), we have</p>
<div class="displaymath process-math" data-contains-math-knowls="./knowl/eq1_8.html">
\begin{equation*}
k=\frac{rS(t)}{e^{rt}-1}.
\end{equation*}
</div>
<p class="continuation">Then according to the problem, we have <span class="process-math">\(r=0.06\)</span> and <span class="process-math">\(t=40\)</span> and <span class="process-math">\(S(t)=8,000,000\text{,}\)</span> therefore,</p>
<div class="displaymath process-math" data-contains-math-knowls="./knowl/eq1_8.html">
\begin{equation*}
k=\frac{0.06 \times 8,000,000}{e^{0.06 \times 40}-1}=\$ 47,889~ \mathrm{year}^{-1}.
\end{equation*}
</div>
<p class="continuation">To save approximately eight million at retirement, the worker would need to save about 50,000 per year over his/her working life. Note that the amount saved over the worker’s life is approximately <span class="process-math">\(40 \times 50,000 =2,000,000\text{,}\)</span> while the amount earned on the investment (at the assumed <span class="process-math">\(6\%\)</span> real return) is approximately <span class="process-math">\(8,000,000-2,000,000=6,000,000\text{.}\)</span> The amount earned from the investment is about <span class="process-math">\(3 \times\)</span>the amount saved, even with the modest real return of <span class="process-math">\(6\%\text{.}\)</span> Sound investment planning is well worth the effort.</p>
<span class="incontext"><a href="sec_4-intro.html#p-14" class="internal">in-context</a></span>
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